Question

Let $ f: \mathbb{R} \to \mathbb{R} $ be a twice differentiable function such that $$ f''(x)\sin\left(\frac{x}{2}\right) + f'(2x - 2y) = (\cos x)\sin(y + 2x) + f(2x - 2y) $$ for all $ x, y \in \mathbb{R} $. If $ f(0) = 1 $, then the value of $ 24f^{(4)}\left(\frac{5\pi}{3}\right) $ is:

• A. \(\left(\frac{5\pi}{3}\right) \) is:
• B. –3
• C. 2
• D. 1

Hint:

When higher derivatives are involved, look for patterns in trigonometric identities and test values at standard angles.

Answer 1

The Correct Option is B

Using substitution and matching with function properties, the fourth derivative is modeled as: \[ f^{(4)}(x) = -\frac{1}{4} \sin \left( \frac{x}{2} \right) \] Then: \[ 24f^{(4)}\left(\frac{5\pi}{3}\right) = 24 \cdot \left( -\frac{1}{4} \cdot \sin\left(\frac{5\pi}{6}\right) \right) = 24 \cdot \left( -\frac{1}{4} \cdot \frac{1}{2} \right) = -3 \]